TRANSACTIONS OF SECTION A. 391 
8. Imaginary Geometry of the Conic. 
By Professor Eutery W. Davis, Ph.D. 
In this paper a comp!ste representation is given of the elements of the 
central conic whose axes are non-similar complex quantities, making it 
depend upon the representation of two auxiliary circles having each one 
of the axes for their radii, The entire plane, with the exception of the 
interior of a certain fundamental conic, is covered with pairs of conics, 
each pair connected by vectors that are the representatives of the complex 
elements, 
9. On the Invention of the Slide Rule. By Professor FLortan Casort. 
When and by whom was the slide rule invented ? Some say Edmund 
Gunter, 1620 or 1624; Augustus De Morgan said William Oughtred, 1632; 
most writers of to-day say Edmund Wingate, 1624, 1626, 1627, or 1630. 
De Morgan’s claim that Gunter invented ‘Gunter’s scale,’ but not the 
slide rule, can be verified by consulting Gunter’s works, which are easily 
accessible. No one denies that Oughtred invented the circular and the 
rectilinear slide rule; that his Latin MS., describing them, was trans- 
lated by William Forster into English and published in 1632 and 1633. 
The main question is, Did Wingate invent the rectilinear slide rule, and 
is he entitled to priority? Hutton, Benoit, Hammer, Favaro, Mehmke, 
D’Ocagne, Henrici say Yes; De Morgan says No. But De Morgan had 
not seen all Wingate’s works, most of the other writers had not seen any. 
We have had examined all of Wingate’s mathematical works, which had 
not been seen previously by De Morgan and Benoit. None contain the 
slide rule, and Oughtred is the inventor of it. The following books by 
Wingate have been examined: L’usage de la regle de proportion en 
Varithmétique et géométrie, 1624, seen by Benoit, copies in Bodleian 
Library (Oxford), in Bibliothéque Nationale and Bibliothéque Mazarine 
(Paris) ; Use of the Rule of Proportion, 1626, 1628, 1645, 1658, 1683, the 
1645 edition seen by De Morgan, copy in British Museum ; Arithmétique 
Logarithmique, 1626, seen by De Morgan; Construction and Use of Line 
of Proportion, 1628, copy in British Museum ; Of Naturall and Artificiall 
Arithmetique, 1630, 1652, copy of 1630 edition in Bodleian, of 1652 edition 
in British Museum; Ludus Mathematicus, 1654, 1681, seen by De Morgan ; 
Use of the Gauge-rod, 2nd edition, in Bodleian ; The Clarks Tutor, 1671, 
1676, copies in Bodleian, 
10. The Asymptotic Expansions of Legendre Functions. 
By J. W. Nicuouson, Nie AS NT) Se* 
The results given in this paper are generalised forms of those relating to 
Bessel functions. The Legendre functions are defined by the contour integrals, 
in the plane of a variable ¢. 
w+,1+,u-, 1-. 
a = Mace o(m +2) 2] jim] o—n (#2 n ae 
Pum) 4n sinner (n) et) (2 (=1)@—p)* 
= Tip l 
1 em tmnt)) m+n 2 x 
Q”,(4) a Aataae ar aoe ) (ni [ore —1)"(t-p)yr-m 
with a cross-cut from ¢=1 to f= —o. 
When m is a positive integer, this definition yields 
Gee. Q,”") ( Bp) J (? Fs 1) q™ | dp™ i aA Q,) (1) 
1 B.A, Report, Dublin, 1908, 
